All language subtitles for Calculus 10_ Modeling with & Solving Differential Equations (Part 1) LM3 Jennifer (Jenny) Wexler-en

JENNIFER WEXLER: Last time we looked at differential equations

from almost entirely an analytic perspective,

and we found solutions to some differential equations,

looked a little bit at their graphical solutions,

but focused primarily on the analytic anti-differentiation pieces.

This time we're going to look almost exclusively

at a graphical analysis of differential equations using a tool

called slope fields.

So slope fields, as you might think from the name,

are quite literally fields of slopes.

So I'm going to put up my field- here is my field,

the coordinate plane- and the little dots

are going to help guide me as to where I'm

going to draw the particular little slopes for the differential equation

that I'm going to create a picture for.

And the differential equation that we're going to start with

is dy dx equals x plus 1.

That's a differential equation we've seen before.

In fact, it was one of the ones we solved analytically last time.

This time I'm going to create a picture on this field of all these little slope

segments, because I want to keep in mind that what dy dx is really telling

me is a slope of a tangent line.

And so I'm just going to pick a point, I think I'll start with 0,2,

and I'm going to look at the differential equation

and say what does that differential equation tell me

about the slope at 0,2?

Well, the slope is x plus 1.

So that's 0 plus 1.

So that's 1.

So to help me visualize that, at the point 0,2

I'm going to draw a little segment of slope approximately 1.

That would be the slope of a line tangent

to a solution curve right there.

If I picked another point, say dy dx at 0,3,

since this differential equation does not depend on y

it doesn't matter that I changed the y to 3.

The slope at that point would still be 1.

And I could draw a little parallel segment at 0,3

showing that a tangent line there would also have a slope of 1.

And in fact anywhere I go on the y-axis I

would see these little slope segments of slope 1.

If I moved to another x-coordinate, say the coordinate negative 1, something.

Again, it doesn't matter what the y-coordinate is,

and so all I care is that the x is negative 1, negative 1 plus 1 is 0.

So if I go to points where the x-coordinate is negative 1- that's

along this column right here, this vertical line-

I would have horizontal slope segments representing slopes of tangent lines

at that particular x value.

And I could continue on to fill in slope segments at all of these points,

and I'm going to take a moment to do so and maybe give you

a chance to do so as well.

And so here we have a more complete slope

field that shows not only the segments that we had already seen together,

with slopes of one and slopes of zero, but I

can see that as the x is increased- as x gets bigger here

the slopes will get steeper.

I see some increasingly steep slope segments.

And as x becomes more and more negative, I

see some increasingly steep negatively sloped segments.

And as we had discussed before, along any vertical line,

since the differential equation only depends on x, all of those little slope

segments are parallel.

And I could try to follow and find a specific solution

curve- you may recall those solution curves were parabolas- I would start

somewhere and I would follow that little slope until I got to the next slope

segment, and continue to try and follow the trend.

And I could see in that slope field, one of the parabolas

that was a particular solution to our differential equation

that we had seen before.

Slope fields are a great way to visualize solutions,

but it's a lot easier to see the solution

when we use some computer generated slope fields, because we'll

get to see more little slope segments coming together.

And so we will now look at not only this particular slope

field created by a computer, but a second slope field,

and do a little bit of a comparison of the two.

So here we have the slope field that we just created by hand:

dy dx equals x plus 1.

So that's the slope field associated with this differential equation,

with the little parallel segments along vertical lines.

Here we have a slope field that's showing a different trend.

It's showing little parallel segments across horizontal lines.

And so, just like these parallel segments

told us that this differential equation depended only

on x, these parallel segments tell me that the differential equation depends

only on y.

That I can change x as I move left and right along the slope field and nothing

happens to my little slope segments.

This differential equation for this slope field

is also one we've seen before.

dy dx equals y.

We saw that last time.

We know from last time that those solution curves

are exponential functions, and if you follow

the curves-- the slope segments to find a curve,

you can see an exponential function, potentially in that slope field.

Here's another one if I start here, there's another exponential function

that I'm seeing in the slope field.

And right along here, along the x-axis, we

see these solutions that are all horizontal slope segments.

All of these points have a y-coordinate of 0,

and so the slopes all have to be 0.

And we saw that solution last time as well, y

equals 0 is a solution curve for this particular differential equation.

So some differential equations depend only

on x, some differential equations depend only on y.

And we're going to now look at one that depends on both x and y

So in this differential equation, we see that the derivative depends

on both x and y, we see an x and a y on the right hand side of the equation.

And so if I were to choose a point, for example the 0.22,

then I can find the slope at that point.

I chose a point where the x and the y-coordinates were equal,

and I see that that gives me a horizontal slope for my tangent line,

and that would be true wherever I have x equals y.

And so along that diagonal I see a number of horizontal tangent lines.

If I were to pick a point where the x was bigger than the y, say 3 comma 2,

then 3 minus 2 is 1, and I would have 3,2 a little 45 degree angle

tangent line.

And everywhere in this area I would have positively sloped segments.

If instead I were to pick a point that, where the x was less than the y,

like 2 comma 3.

Now I have a slope of negative 1.

So if I go to the point 2 comma 3 my slope is negative 1.

And everywhere up here, where the x is less than y,

I would have negatively sloped segments.

So if you want some practice you could take some time to try and create

the rest of this slope field by hand.

I'm going to move to a computer generated version.

So here we have a computer generated version of the slope field

that we were just working on by hand.

The slope field for the differential equation dy dx equals x minus y.

And I can see in this slope field the little horizontal slopes at 1,1,

right all along the diagonal where y equals x.

You'll notice in the computer generated slope field I'm not

stuck with just these integer points.

I'm allowed to see a little more detail at other points.

But I do see those horizontal segments exactly where

I would expect to see them.

I do see those positively sloped segments

exactly where I expect to see them.

And I do see those negatively sloped segments

exactly where expect to see them.

And you can see there are no-- if I pick any vertical line,

those slope segments are not all parallel.

So I know that this has both an x and a y in it; same thing for a horizontal,

right?

There's no parallel slope segments along horizontal and vertical lines.

And so I know that my differential equation

must have a derivative that depends on both coordinates.

So I'm now going to move to a dynamic graph

so we can look at some of the particular solutions

within this field of slopes that shows us the general solution.

So now we have this graph right here, that's

the same slope field that we were just looking at for dy dx equals x minus y.

But in addition to the slope field, there's

a particular solution sketched in.

It happens to be the particular solution that

contains the 0.01 that's a solution to this differential equation

that we've seen previously.

Which solution curve appears depends on what point I know the curve contained.

So now I have a very different looking solution curve

that goes to the 0.4,0.5 negative 2.4.

If I were to change that point again, now that's

the solution curve for the curve that contains negative 1.3, negative 2.6.

And you may notice, as I drag this around,

that the curves have many different shapes.

There's one that's a straight line, that is a solution to this differential

equation.

They are all related, they are all part of the same family of solutions,

and changing that known condition, that point on the curve,

allows us to see the different variations in the family.

So before moving on you'll have a chance to create a slope field by hand,

and you'll also have a chance to look at some computer generated slope fields

and interpret the information found in them.